We regress to a predicted rotation \(\hat{\mathbf {r}}\)  represented in the axis-angle form.
The prediction is compared against the ground truth rotation \(\mathbf {r}\)  via a loss function \(l:\mathbb {R}^3\times \mathbb {R}^3\rightarrow \mathbb {R}_{\ge 0}\) .
Let \(\hat{R}\)  and \(R\)  denote the two rotation matrices corresponding to \(\hat{\mathbf {r}}\)  and \(\mathbf {r}\) , respectively.
We use as loss function the geodesic distance \(d(\hat{R}, R)\)  of \(\hat{R}\)  and \(R\)  [1]}, [2]}, i.e.,
\(l(\hat{\mathbf {r}}, \mathbf {r}) = d(\hat{R}, R) = \phi (\hat{R}R^T),\) 
